Determine the value of the following complex number power. Your answer will be plotted in orange. $ ({\cos(\frac{3}{4}\pi) + i \sin(\frac{3}{4}\pi)}) ^ {5} $
Let's express our complex number in Euler form first. $ {\cos(\frac{3}{4}\pi) + i \sin(\frac{3}{4}\pi)} = { e^{3\pi i / 4}} $ Since $(a ^ b) ^ c = a ^ {b \cdot c}$ $ ({ e^{3\pi i / 4}}) ^ {5} = e ^ {5 \cdot (3\pi i / 4)} $ The angle of the result is $5 \cdot \frac{3}{4}\pi$ , which is $\frac{15}{4}\pi$ Our result is $ e^{7\pi i / 4}$. Converting this back from Euler form, we get $\cos(\frac{7}{4}\pi) + i \sin(\frac{7}{4}\pi)$.